D₁₁₉₈₅ #37561798

order	23970
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₉₈₅
unambiguous semidirect product	C₃⋊D₃₉₉₅
unambiguous semidirect product	C₅⋊D₂₃₉₇
unambiguous semidirect product	C₁₇⋊D₇₀₅
unambiguous semidirect product	C₄₇⋊D₂₅₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₇×C₄₇ ⋊ C₂
Hall	C₅×C₁₇×C₄₇ ⋊ D₃
Hall	C₃×C₁₇×C₄₇ ⋊ D₅
Hall	C₁₇×C₄₇ ⋊ D₁₅
Hall	C₃×C₅×C₄₇ ⋊ D₁₇
Hall	C₃×C₅×C₁₇ ⋊ D₄₇
Hall	C₅×C₄₇ ⋊ D₅₁
Hall	C₃×C₄₇ ⋊ D₈₅
Hall	C₅×C₁₇ ⋊ D₁₄₁
Hall	C₃×C₁₇ ⋊ D₂₃₅
Sylow	C₄₇ ⋊ D₂₅₅
Sylow	C₁₇ ⋊ D₇₀₅
Hall	C₃×C₅ ⋊ D₇₉₉
Sylow	C₅ ⋊ D₂₃₉₇
Sylow	C₃ ⋊ D₃₉₉₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₇×C₄₇	C₂
LCS	G ▷ C₃×C₅×C₁₇×C₄₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 11985

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72);(2,3)(5,8)(6,7)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2,3,5,17,47|b^a=b^2,c^a=c^4,d^a=d^16,e^a=e^46

Conjugacy classes & character table

5994 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23970, 16);

D11985 #37561798